Symplectic Floer homology of area-preserving surface diffeomorphisms

Abstract

The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class.Comment: 57 pages, 4 figures. Revision for publication, with various minor corrections. Adds results on the module structure and invariance thereo

The growth rate of symplectic Floer homology

“…We give now, following Cotton-Clay [4], a notion of weak monotonicity such that HF * (φ) is well defined and invariant among weakly monotone symplectomorphisms. Monotonicity implies weak monotonicity, and so HF * (g) = HF * (φ) for any weakly monotone φ in a mapping class g. The properties of weak monotone symplectomorphism of a surface play a crucial role in the computation of Floer homology for pseudo-Anosov and reducible mapping classes (see [4]).…”

“…Monotonicity implies weak monotonicity, and so HF * (g) = HF * (φ) for any weakly monotone φ in a mapping class g. The properties of weak monotone symplectomorphism of a surface play a crucial role in the computation of Floer homology for pseudo-Anosov and reducible mapping classes (see [4]). Fundamental properties of monotone symplectomorphisms listed above are also satisfied when replacing monotonicity with weak monotonicity.…”

“…Seidel [35] and CottonClay [4] have proved that if φ is monotone or weakly monotone, then the energy is constant on each M k (y − , y + ; J, φ). Since all fixed points of φ are nondegenerate, the set Fix(φ) is a finite set and the Z 2 -vector space…”

“…In this section, we review standard form maps as discussed in [19,4]. These are special representatives of mapping classes adopted to the symplectic geometry.…”

“…Every fixed point is in a separate Nielsen class and each of the Nielsen classes, for which there is a fixed point, has index +1. For a pseudo-Anosov mapping class, a standard form map is a symplectic smoothing (see [4]) of the singularities and boundary components of the canonical singular representative. Each singularity has a number p ≥ 3 of prongs and each boundary component has a number p ≥ 1 of prongs.…”